Halin [1]} provides an alternative characterization of tree-width as a maximal element in a class of functions called S-functions. These are mappings from finite graphs to \(\mathbb {N}\)  satisfying a set of common properties fulfilled by the chromatic number, vertex-connectivity number and the Hadwiger number. In order to define S-functions, we first recall the concept of an \(H\) -sum of two graphs.
